Calculus application of differentiation brain buster ughhh...help?

Question

ok i cant seem to set this problem up correctly...
A fence 8ft tall runs parallel to a tall building at a distance 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fenceto the wall of the building?

so what i am trying to do is get the minimum value of the equation i am trying to set up...

Answer 1

The shortest ladder will be the hypotenuse of the isoceles right -angled triangle which is 8' high, 4' from the vertical. If I remember my trig, this means that the triangle formed by the ladder and the fence, and the ladder and the wall are similar, so that the base of the ladder/fence triangle is 8'. As the fence is 4' from the wall, the base is therefore 12'. As the triangle is isoceles, the height is 12'. Therefore the hypotenuse is sqrt(12^2 +12^2) = 12rt(2)' or just under 17'. Where's the calculus? Unless the problem specifically asked for calculus, don't make difficulties for yourself.

Answer 2

This is actually a trickier problem than it looks.
Draw a picture of the situation. Let the angle that the ladder makes with the ground be called A. Then the distance from the ground to the top of the ladder is 8/sin(A) and the distance from the top of the ladder to the wall is 8/cos(A). The total length of the ladder is then
L=8csc(A)+4sec(A).

The derivative of this with respect to A is then
dL/dA=-8csc(A)cot(A)+
4sec(A)tan(A).

Set this equal to 0, and reatrrange to get
tan^3 (A)=8/4=2,
so tan(A)=cbrt(2) [not sqrt(2) like Alberd says].
When you draw the triangle for the angle A, you will find that
sec(A)=sqrt[1+cbrt(4)] and
csc(A)=
sqrt[1+cbrt(4)]/cbrt(2).

When you plug these back into the formula for the length, L, you can simplify this and find that the minimal length is

L=4[1+cbrt(4)]^(3/2)

Answer 3

Let x be the distance from the foot of the ladder to the top of the fence. Then sqrt(x^2-64) = distance fro foot of ladder to the fence, and sqrt(x^2-64)+4 = distance from foot of ladder to building. Let the foot of the ladder be at point A annd the end of the ladder at the building = B. AB is length of the ladder

Then x/sqrt(x^2-64) = AB/[sqrt(x^2-64)+4]
Now solve for AB and find dAB/dx, set it to 0, and solve for x. You will then be able to compute length of ladder.

The easiest way is to realize that the minimum length will occur when the ladder has an angle of inclination of 45 degrees. This give AB = ladder length = 12sqrt(2)

Answer 4

Set up the equations: y = length of the ladder, z = height of building where the ladder touches, x = the distance of the foot of the ladder from the fence.

Use similar triangles to set up an equation between z and x.

Set up an equation with y and x. Differentiate, set to zero, and solve.

I get 16.65

Answer 5

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